{"id":"cb_question_bank:b8a3b472-476e-4b02-8ee3-147abc15ccea","source_id":"cb_question_bank","source_item_id":"b8a3b472-476e-4b02-8ee3-147abc15ccea","external_id":"5c9efc7d-a1db-4027-b9b5-a10ca2abedcd","ibn":null,"question_id":"f243c383","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.A.","skill":"Area and volume","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Two identical rectangular prisms each have a height of 90 centimeters left parenthesis cm right parenthesis . The base of each prism is a square, and the surface area of each prism is upper K cm squared . If the prisms are glued together along a square base, the resulting prism has a surface area of StartFraction 92 Over 47 EndFraction upper K cm squared . What is the side length, in cm , of each square base?","stimulus":"","stimulus_html":null,"stem_html":"

Two identical rectangular prisms each have a height of 90 centimeters cm. The base of each prism is a square, and the surface area of each prism is K cm2. If the prisms are glued together along a square base, the resulting prism has a surface area of 9247K cm2. What is the side length, in cm, of each square base?

","rationale_html":"

Choice B is correct. Let x\n represent the side length, in cm, of each square base. If the two prisms are glued together along a square base, the resulting prism has a surface area equal to twice the surface area of one of the prisms, minus the area of the two square bases that are being glued together, which yields 2K-2x2 cm2 . It’s given that this resulting surface area is equal to 9247K cm2, so 2K-2x2=9247K. Subtracting 9247K from both sides of this equation yields 2K-9247K-2x2=0. This equation can be rewritten by multiplying \n\t2\n\tK\n\n on the left-hand side by 4747, which yields 9447K-9247K-2x2=0, or 247K-2x2=0. Adding 2x2 to both sides of this equation yields 247K=2x2. Multiplying both sides of this equation by 472 yields K=47x2. The surface area K\n, in cm2, of each rectangular prism is equivalent to the sum of the areas of the two square bases and the areas of the four lateral faces. Since the height of each rectangular prism is 90 cm and the side length of each square base is x cm, it follows that the area of each square base is x2 cm2 and the area of each lateral face is 90x cm2. Therefore, the surface area of each rectangular prism can be represented by the expression 2x2+490x, or 2x2+360x. Substituting this expression for K\n in the equation K=47x2 yields 2x2+360x=47x2. Subtracting 2x2 and \n\t360\n\tx\n\n from both sides of this equation yields 0=45x2-360x. Factoring x\n from the right-hand side of this equation yields 0=x45x-360. Applying the zero product property, it follows that x=0 and 45x-360=0. Adding 360\n to both sides of the equation 45x-360=0 yields 45x=360. Dividing both sides of this equation by 45\n yields x=8. Since a side length of a rectangular prism can’t be 0\n, the length of each square base is 8\n cm.

\n

Choice A is incorrect and may result from conceptual or calculation errors.

\n

Choice C is incorrect and may result from conceptual or calculation errors.

\n

Choice D is incorrect and may result from conceptual or calculation errors.

","correct_answer":["B"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.836000+00:00","source_updated_at":"2023-08-02T20:25:59.836000+00:00","options":[{"label":"A","content_html":"

4\n

","ord":0},{"label":"B","content_html":"

8\n

","ord":1},{"label":"C","content_html":"

9\n

","ord":2},{"label":"D","content_html":"

16\n

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